Out of the flurry of ambitious gun control proposals in the wake of December’s school shooting in Connecticut, expanded background checks on gun sales are fast emerging as the “sweet spot” — as one Senate Democratic leader put it — between what gun control advocates seek and what can actually attract bipartisan support in Congress.

Discussion of bans on so-called assault weapons and high-capacity ammunition magazines have gradually given way to the more politically palatable ideas of strengthening the federal background check system or instituting near-universal background checks for gun buyers — even as proponents and opponents clash bitterly over the merits of such measures.

Mark Glaze, director of the gun control advocacy group Mayors Against Illegal Guns, said that while they support bans on so-called assault weapons and high-capacity magazine clips, their focus is on background checks.

“That’s our top priority,” he said. “Other organizations may prioritize different things, but I think we agree that that is the biggest problem and also the thing that we can probably do.”

House Majority Leader Eric Cantor said recently that the reforms his state made in the wake of the 2007 Virginia Tech shootings linking mental health records to background check databases could serve as a model for the country.

“I think that we can take a lot of lessons from what Virginia did and put it in place at the federal level, because there are a lot of states that aren’t doing what Virginia is doing to try and beef up the database for the background checks,” the House’s No. 2 Republican said in an interview with CNN.

In the wake of Virginia Tech, Sen. Tim Kaine, then governor of the state, signed an executive order to close a loophole that had allowed people adjudicated mentally ill, such as gunman Seung-Hui Cho, to purchase firearms.

“Nineteen states don’t put any mental health adjudications into the national database, and some states just do a few here and there,” Mr. Kaine said. “But Virginia is the best state in the country in doing that, so there are aspects of the way we do the background check system in Virginia that really are a national model.”

Democratic Sen. Mark R. Warner, who has an A rating from the National Rifle Association, predicted broad bipartisan support for background checks on virtually all gun buyers while providing exemptions for transactions between family members or friends at a shooting range, for example.

“How you look at the vast amount of purchases that are made within the system right now without any appropriate checks, the 19 states who don’t even report those folks who have been involuntarily committed, their mental health records at least getting into the database — these are areas where reasonable people ought to agree and we can at least take a major first step,” he said.

The NRA opposes universal background checks, saying they would contribute to an unnecessarily bloated bureaucracy and serve mostly as a hassle for law-abiding citizens.

The gun rights group, however, has been lobbying to strengthen the National Instant Criminal Background Check System for 20 years by supplementing it with more mental health records, said NRA President David Keene.

That Mr. Cantor, Virginia’s two Democratic senators and the NRA all agree with that principle perhaps makes Virginia’s strategy of beefing up existing databases a true “sweet spot,” the term Sen. Charles E. Schumer, New York Democrat, used to describe universal background checks recently on “Meet the Press.”

In 1993, NRA CEO Wayne LaPierre appeared on CBS’ “Face the Nation” with Mr. Schumer, then a congressman, Mr. Keene said, and talked about adding more mental health records into the system.

“Wayne made the point that we believe very strongly that those people need to be included in the system of background checks,” he said. “He and Charles Schumer shook hands and Schumer said, ‘Wayne, I can help you do that.’”

But John Lott, a prominent author who studies the link between gun laws and gun crime, said tighter background checks wouldn’t have stopped the Newtown shootings.

Adam Lanza allegedly took guns from his mother and fatally shot her before killing 20 children and six adults at Sandy Hook Elementary School.

“They keep on going back to this background check stuff, but it doesn’t have anything to do [with it],” Mr. Lott said, adding that about 80 percent of the checks take three days. “These types of background checks are problematic. They could virtually all be false positives.”

Some states, including California and Rhode Island, already require universal checks on all purchases, regardless of the type of weapon or venue. But gun control advocates say 40 percent of gun sales are done without background checks because of a loophole that allows private dealers to sell firearms without doing checks.

Mr. Keene and other gun rights advocates say the number is more in the range of 4 percent to 20 percent, and that the higher number is based on a flawed, decades-old telephone survey.

Regardless of those figures, background checks on all gun sales enjoy near-universal support from voters in recent polls. A Quinnipiac University poll released Thursday showed that 92 percent of registered voters support background checks for all gun sales; 56 percent support bans on so-called assault weapons and high-capacity magazine clips; and a plurality — 46 percent to 43 percent — say the NRA better reflects their views on guns compared to Mr. Obama.

“You know, existing laws say if you’re a felon, if you’re a fugitive from justice, if you’ve been adjudicated mentally ill and dangerous, domestic violence abuser under protective order, you cannot purchase a weapon,” Mr. Kaine said. “The only way to enforce that is to have [a] background check, so we either want to enforce those laws or we don’t. Background checks help enforce those laws, and that’s why they’re so popular.”